# Thread: Subspaces Projection and R^n

1. ## Subspaces Projection and R^n

Two questions am not understanding?
1) It says;

U1={X belong to R^n:AX=3X}

and

2)

We fix a non-zero vector d belonging to R^3, and define U3={X beloninig to R^3: proj(d)(X)=0}, where projd(x) is the projection of X onto d

2. Hi, what is A in 1). Is it a matrix, linear map, ..? If yes, the answer is positive and you need only to show that for any $X,Y\in U_1\ and\ c\in\mathbb{R},\ then\ X+Y\in U_1\ and\ cX\in U_1$
you can use the same argument for 2)

3. It is a matrix...am not following ur logic? please expand on it if you may i have more trouble with the projection

4. To show that S is a subspace of $\mathbb{R}^n$, it suffice to show it's stable by addition and multiplication by a scalar (real number). So you need to prove that $U_1$ has these properties using the fact that $A(X+Y) = AX+AY$ and $A(cX) = cAX$ i.e. A is linear. Also multiplication by 3 is linear.

for 2), proj(d) is also linear.

5. okay i understoood the first part as it is a subspace so itrs true, am still not sure how to show the proj one? i was never to well with projectin?please help

6. Projecting of X on a vector d is equivalent to taking the scalar product X.d and form the vector (X.d)d/(d^2) i.e. proj(d)(X) = (X.d)d/(d^2) (d^2 is the square of the norm of d). You should know that scalar product is linear i.e. (X+Y).d = X.d+Y.d and (cX).d = c(X.d)